Day 6 – May 16 – WBL 3.3-3.4 The motion of an object that is subjected to a constant, downward acceleration is termed projectile motion. We already covered.

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Presentation on theme: "Day 6 – May 16 – WBL 3.3-3.4 The motion of an object that is subjected to a constant, downward acceleration is termed projectile motion. We already covered."— Presentation transcript:

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Day 6 – May 16 – WBL The motion of an object that is subjected to a constant, downward acceleration is termed projectile motion. We already covered 1D projectile motion in chapter 2 – there, objects were either dropped or thrown vertically. The constant acceleration (a = -g) dictated the position of the object over time. Now, we will allow the object to move in two dimensions (why not 3?) As we saw in the last lecture, we can break up this 2D problem into two 1D problems, for the horizontal (x) and vertical (y) directions. By combining these two solutions, we can determine the path over which the object travels. The relevant equations are repeated here (originally from slide 9 of the last lecture): 3.3 Projectile Motion PC141 Intersession 2013Slide 1

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Day 6 – May 16 – WBL Since the acceleration is downward, we can state that a y = -g and a x = 0. In other words, the horizontal motion is defined by zero acceleration (constant velocity), and the vertical motion is defined by constant acceleration. 3.3 Projectile Motion PC141 Intersession 2013Slide 2 Horizontal Projection We begin with the case that the projectiles initial velocity is horizontal. Since there is no horizontal acceleration, we can say that at all times, v x = v x0. Furthermore, this means that the horizontal position is a linear function of time: x = x 0 + v x0 t.

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Day 6 – May 16 – WBL While the projectile is traveling at a constant horizontal velocity, it is also traveling vertically. However, since there is a non-zero vertical acceleration, the vertical velocity is not constant. This causes the projectile to move in a curved path. 3.3 Projectile Motion PC141 Intersession 2013Slide 3 The figure to the right shows two objects that are released at the same time – the red ball is dropped from rest while the yellow ball is projected horizontally. At any given time, both balls have dropped the same vertical distance. The vertical motion and horizontal motion are uncoupled (they dont depend on each other). Both balls will land at exactly the same time.

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Day 6 – May 16 – WBL At this point, the text proclaims that the curve described by these equations…is called a parabola. Lets prove that, since its not too hard. The previous equations describe x and y as functions of time. However, the figure on slide 4 shows the trajectory of the projectile, which is actually y as a function of x. Nothing is plotted as a function of time (although the position of the object is shown at 7 particular times during the projectiles flight). To find y(x), we follow two steps: 1.Invert the x(t) equation to find t(x) 2.Substitute this into the y(t) equation to convert it to y(x) 3.3 Projectile Motion PC141 Intersession 2013Slide 7

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Day 6 – May 16 – WBL Problem #1: Thrown Football PC141 Intersession 2013Slide 9 A football is thrown on a long pass. Compared to the balls initial horizontal velocity component, the magnitude of the velocity at the highest point is… A greater B less C the same WBL LP 3.9 (modified)

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Day 6 – May 16 – WBL Problem #2: Kicked Football PC141 Intersession 2013Slide 13 The figure shows three paths for a football kicked from ground level. Note that all three paths have the same maximum height, but they have different ranges. a)Rank them in terms of time of flight (longest to shortest) b)Rank them in terms of initial vertical velocity component (fastest to slowest) c)Rank them in terms of initial horizontal velocity component (fastest to slowest) d)Rank them in terms of initial speed (fastest to slowest) Solution: In class

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Day 6 – May 16 – WBL Problem #3: Long Jump PC141 Intersession 2013Slide 14 In the 1991 World Track and Field Championships, Mike Powell jumped a distance of 8.95 m (breaking a 23-year-old world record by 5 cm). Assume that Powells speed on takeoff was 9.5 m/s. a)How much less was Powells range than the maximum possible range for an object launched at the same speed? b)What is the greatest source of the discrepancy in a)? c)What would Powells distance have been if he had run just a bit faster, with a speed on takeoff of 9.6 m/s (and the same initial angle as before)? Solution: In class

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Day 6 – May 16 – WBL Problem #4: Flying Fish PC141 Intersession 2013Slide 15 Fish use various techniques to avoid their predators. The California flying fish propels itself out of the water using its tail, at a typical speed of 30 km/h. a)If this fish left the water at a 45° angle, how far could it travel through the air? b)Suppose a nearby fisherman measured the distance as 180 m. How could this be explained? Solution: In class

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Day 6 – May 16 – WBL Problem #5: Volleyball PC141 Intersession 2013Slide 16 A volleyball court is 18.0 m long (end-to-end), and the top of the net is 2.24 m above the ground (for womens competition). Using a jump serve, a player strikes the ball at a point that is 3.00 m above the ground, and 8.00 m away from the net. The initial motion of the ball is horizontal. a)What is the minimum initial velocity if the ball is to clear the net? b)What is the maximum initial velocity of the ball is to land in bounds on the other side of the net? Solution: In class

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Day 6 – May 16 – WBL In chapter 2, we learned that an objects position is measured relative to an arbitrary origin. As it turns out, any measurement of an objects velocity is also dependent on the velocity of the observer. As an example, you may think youre jogging at 5 m/s, but thats only relative to a reference frame that is pinned to the Earth. Since the Earth is spinning about its axis, and rotating about the sun (which itself is moving relative to the center of the Milky Way galaxy), an alien who is observing you from far away would claim that youre moving much faster than 5 m/s. In most cases in PC141, the pinned to the Earth reference frame is appropriate. However, sometimes we need to deal with velocities relative to each other. 3.4 Relative Velocity PC141 Intersession 2013Slide 17

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Day 6 – May 16 – WBL Problem #7: Airplane with Crosswind PC141 Intersession 2013Slide 22 An airplane is flying at 150 km/h (its speed in still air) in a direction such that with a wind of 60 km/h blowing from east to west, the airplane travels in a straight line southward. a)What must be the planes heading (direction) for it to fly directly south? b)If the plane has to go 200 km in the southward direction, how long does it take? Solution: In class WBL Ex 3.84 (modified)